No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
You would be forgoing your rights under the identity theft provisions of the FCRA by settling the debt.
By simply sending a copy of the police report to the CRA, you can get the information related to the identity theft blocked from your CR. FCRA 605B.
You can additionally send a copy of the police report to the credtior, and demand under the provisions of FCRA 609(e) a copy of all business records in their possession relating to the unauthorized use of your account or any transactions made without your authorization.
If, having been advised of the identity theft, they fail to hold that you are not responsible for the debt, you can use the information obtained from them under the provisions of section 609(e) to initiate civil litigation. Check out your state statutes for protections upon notification of a creditor of identity theft/unauthorized use of your account. Your state consumer protection bureau would be a good place for advice. It is not your debt.
I would not advise paying the debt.